On negative outcome control of unobserved confounding as a generalization of difference-in-differences

The difference-in-differences (DID) approach is a well known strategy for estimating the effect of an exposure in the presence of unobserved confounding. The approach is most commonly used when pre-and post-exposure outcome measurements are available, and one can assume that the association of the unobserved confounder with the outcome is equal in the two exposure groups, and constant over time. Then, one recovers the treatment effect by regressing the change in outcome over time on the exposure. In this paper, we interpret the difference-in-differences as a negative outcome control (NOC) approach. We show that the pre-exposure outcome is a negative control outcome, as it cannot be influenced by the subsequent exposure, and it is affected by both observed and unobserved confounders of the exposure-outcome association of interest. The relation between DID and NOC provides simple conditions under which negative control outcomes can be used to detect and correct for confounding bias. However, for general negative control outcomes, the DID-like assumption may be overly restrictive and rarely credible, because it requires that both the outcome of interest and the control outcome are measured on the same scale. Thus, we present a scale-invariant generalization of the DID that may be used in broader NOC contexts. The proposed approach is demonstrated in simulations and on a Normative Aging Study data set, in which Body Mass Index is used for NOC of the relationship between air pollution and inflammatory outcomes.